Are Convolutional Neural Networks better than existing image recognition libraries that don't use CNNs?
It would not be wise to say that CNNs are better objectively than traditional approaches to solve computer vision problems as there are many problems for which the traditional methods works just fine. CNNs do have an inherent advantage over traditional methods which is the same advantage that deep learning has over other traditional methods i.e learning hierarchical features i.e what features are useful and how to compute them.

The traditional way to approach a CV problem is to figure out the features that are relevant to the problem, figure out how to compute those features and then use those features to compute the final result. Whereas in CNN case the training process will figure out all the 3 points for you given that you have huge number of training examples.